Young children need calcium in their diet to support the growth of their bones.

Question

Young children need calcium in their diet to support the growth of their bones. The Institute of Medicine provides guidelines for how much calcium should be consumed by people of different ages. One study examined whether or not a sample of children consumed an adequate amount of calcium based on these guidelines. Since there are different guidelines for children aged 5 to 10 years and those aged 11 to 13 years, the children were classified into these two age groups. Each student's calcium intake was classified as meeting or not meeting the guideline. There were 2,028 children in the study. Here are the data.

                                             Age (years)
Met requirement                  5 to 10   11 to 13
No                                          197         552
Yes                                        865         414

Use a significance test to make the comparison.

a.) State the null and alternative hypotheses. (choose one option below) (answers options are in pairs)
H0: page 5-10 page 11-13
Ha: page 5-10 < page 11-13

H0: page 5-10 = page 11-13
Ha: page 5-10 page 11-13    

H0: page 5-10 > page 11-13
Ha: page 5-10 page 11-13

H0: page 5-10 page 11-13
Ha: page 5-10 > page 11-13

b.) Report the test statistic and the P-value. (Use page 5-10 page 11-13. Round your value for z to two decimal places and your P-value to four decimal places.)

z =
P-value =

c.) Interpret the result of your test. (Use = 0.05.) (choose one option)

Reject the null hypothesis. There is not significant evidence that page 5-10 is different from page 11-13.

Reject the null hypothesis. There is significant evidence that page 5-10 is different from page 11-13.    

Fail to reject the null hypothesis. There is significant evidence that page 5-10 is different from page 11-13.

Fail to reject the null hypothesis. There is not significant evidence that page 5-10 is different from page 11-13.

d.) Justify for the use of the large-sample procedure for this comparison. (choose one option)

The data are large simple random samples from two independent populations.

The data are large simple random samples from two dependent populations.    

The data are small simple random samples from two independent populations.

The data are small simple random samples from two dependent populations.

Explanation / Answer

Solution:

page 5-10 = 865/1062 = 0.8145

nage 5-10 = 1062

page 11-13 = 414/966 = 0.42857

nage 11-13 = 966

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: Page 5-10 = Page 11-13

Alternative hypothesis: Page 5-10 Page 11-13

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the proportion from population 1 is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p1 * n1 + p2 * n2) / (n1 + n2)

p = 0.63067

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

SE = 0.02146

z = (p1 - p2) / SE

z = 17.98

where p1 is the sample proportion in sample 1, where p2 is the sample proportion in sample 2, n1 is the size of sample 1, and n2 is the size of sample 2.

Thus, the P-value = 0.00001

Interpret results. Since the P-value (0.00001) is less than the significance level (0.05), we cannot accept the null hypothesis.

From this we can conclude that there is significant evidence that page 5-10 is different from page 11-13.

The data are large simple random samples from two dependent populations.

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